Random recurrent neural networks with delays
This paper is dedicated to study the asymptotic behavior of some class of random functional differential equations of the form \[ \frac{dx_{i}(t,\omega)}{dt}=-x_{i}(t,\omega)+\sum\limits_{j=i-N}^{i+N}\lambda_{ij}(\theta_{t}\omega)F_{i}(t,x_{jt}(\omega))+J_{i}(t),\ i\in\mathbb Z \] with the initial condition \[ x_{i}(t)=\phi_{i}(t-\tau),\ \ t\in [\tau -h,r],\ \ i\in\mathbb Z \] or in the vector form \[ \frac{dx(t,\omega)}{dt}=G(t,\theta_{t}\omega,x_{t})\ \ \mbox{for all}\ \ t\ge \tau,\ \ x(t)=\phi(t-\tau) \ \mbox{for all}\ t\in [\tau-h,\tau],\tag{3} \] where \[ \phi \in l^{2}:=\{x=(x_{i})_{i\in\mathbb Z}:\ \sum\limits_{i\in\mathbb Z}x_{i}^{2}<\infty \},\tag{4} \] \((\Omega,\mathcal F,\mathbb R)\) is a probability space, \((\Omega,\mathbb R,\theta))\) is a random dynamical system on the event space \(\Omega\) and \((Q,\mathbb R,\xi)\) is a deterministic dynamical system on the metric space \(Q\). The authors study the problem \begin{itemize} \item[--] of existence of a solution (at least one) of the Cauchy problem (3) and they give sufficient conditions under which equation (3) generates a non-autonomous random cocycle dynamical system; \item[--] of existence of random pullback attractors for equation (3); \item[--] of periodicity of attractors for periodic equation (3); \item[--] of periodicity of trajectories in the random pullback attractors. \end{itemize} I consider that that these results are new and interesting for the experts in the nonautonomous functional differential equations and random dynamical systems.
multi-valued non compact random dynamical system
scientific article; zbMATH DE number 7216762
scientific article